Course Description

We are interested in the computational aspects
of the geometry and topology of mathematical objects such
as curves and surfaces.
Such objects are usually defined by their metric/local properties
(like position and curvature).
These properties then define
topological invariants (like connectivity or orientability)
which are global in nature.
For instance, a surface S may be specified by
their equation F(X,Y,Z)=0. This equation may be regarded
as a metric/local description of S.
Now, suppose we want to
compute global properties like the number of connected components,
or the topological type of S from this equation.

Invariably, the first step is to compute a combinatorial
approximation of S from the given metric/local information.
This combinatorial approximation is called a mesh (or complex).
This is the meshing problem.
Meshing is a critical step -- as the interface between
continuous and discrete computation. The computation
involves some combination of numerical and algebraic techniques.
A big question is how to guarantee
that this step is correct in the presense of numerical errors.
(Most published algorithms have no guarantees.)
The computational methods depend on the representation
of the object. E.g., we would need very different
algorithms if the surface S in the previous example
were represented by Bezier patches.

The next step is to compute the desired
topological property of S from an appropriate mesh representation.
The mathematical and computational tools such as
homology, Morse theory and subdivision schemes
will be developed.

Books and References

We will cover two chapters from a forthcoming book
Effective Computational Geometry for Curves and Surfaces
(Eds., J.-D.Boissonnat and M.Teillaud):